Fully anonymous rooms aren't a feature in HipChat. Your display name and mention name are defined at the user level and aren't something that can be changed on a per-room basis.
One option if you wanted to emulate this would be to enable guest access and have anyone who joins the room use the guest URL. This would let each person choose a name of their choice when entering the room. However, it has the drawbacks that you can't chat in the desktop client (guest access is web only) and your history won't be associated with your user.
...have mention name FirstnameLastname. Fe.: @MaartenCautreels This certainly helps to know who is being mentioned in a conversation. Imagine your colleague sends the following message...
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